Thiết kế móng thiết bị (Foundations for Dynamic Equipment)

ACI 351.3R-04 became effective May 3, 2004.
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351.3R-1
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Foundations for Dynamic Equipment
ACI 351.3R-04
This report presents to industry practitioners the various design criteria
and methods and procedures of analysis, design, and construction applied
to dynamic equipment foundations.
Keywords: amplitude; concrete; foundation; reinforcement; vibration.
CONTENTS
Chapter 1—Introduction, p. 351.3R-2
1.1—Background
1.2—Purpose
1.3—Scope
1.4—Notation
Chapter 2—Foundation and machine types,
p. 351.3R-4
2.1—General considerations
2.2—Machine types
2.3—Foundation types
Chapter 3—Design criteria, p. 351.3R-7
3.1—Overview of design criteria
3.2—Foundation and equipment loads
3.3—Dynamic soil properties
3.4—Vibration performance criteria
3.5—Concrete performance criteria
3.6—Performance criteria for machine-mounting systems
3.7—Method for estimating inertia forces from multi-
cylinder machines
Chapter 4—Design methods and materials,
p. 351.3R-26

4.1—Overview of design methods
4.2—Impedance provided by the supporting media
4.3—Vibration analysis
4.4—Structural foundation design and materials
4.5—Use of isolation systems
4.6—Repairing and upgrading foundations
4.7—Sample impedance calculations
Chapter 5—Construction considerations,
p. 351.3R-53
5.1—Subsurface preparation and improvement
5.2—Foundation placement tolerances
5.3—Forms and shores
5.4—Sequence of construction and construction joints
5.5—Equipment installation and setting
5.6—Grouting
5.7—Concrete materials
5.8—Quality control
Reported by ACI Committee 351
William L. Bounds
*
Fred G. Louis Abdul Hai Sheikh
William D. Brant Jack Moll Anthony J. Smalley
Shu-jin Fang Ira W. Pearce Philip A. Smith
Shraddhakar Harsh Andrew Rossi
*
W. Tod Sutton
†
Charles S. Hughes Robert L. Rowan, Jr.
‡
F. Alan Wiley

Erick Larson William E. Rushing, Jr.
James P. Lee
*
Chair
Yelena S. Golod
*
Secretary
*
Members of the editorial subcommittee.
†
Chair of subcommittee that prepared this report.
‡
Past chair.
351.3R-2 ACI COMMITTEE REPORT
Chapter 6—References, p. 351.3R-57
6.1—Referenced standards and reports
6.2—Cited references
6.3—Software sources and other references
6.4—Terminology
CHAPTER 1—INTRODUCTION
1.1—Background
Heavy machinery with reciprocating, impacting, or rotating
masses requires a support system that can resist dynamic
forces and the resulting vibrations. When excessive, such
vibrations may be detrimental to the machinery, its support
system, and any operating personnel subjected to them.
Many engineers with varying backgrounds are engaged in
the analysis, design, construction, maintenance, and repair of
machine foundations. Therefore, it is important that the
owner/operator, geotechnical engineer, structural engineer,

and equipment supplier collaborate during the design
process. Each of these participants has inputs and concerns
that are important and should be effectively communicated
with each other, especially considering that machine foundation
design procedures and criteria are not covered in building
codes and national standards. Some firms and individuals
have developed their own standards and specifications as a
result of research and development activities, field studies,
or many years of successful engineering or construction
practices. Unfortunately, most of these standards are not
available to many practitioners. As an engineering aid to
those persons engaged in the design of foundations for
machinery, the committee developed this document, which
presents many current practices for dynamic equipment
foundation engineering and construction.
1.2—Purpose
The committee presents various design criteria and
methods and procedures of analysis, design, and construction
currently applied to dynamic equipment foundations by
industry practitioners.
This document provides general guidance with reference
materials, rather than specifying requirements for adequate
design. Where the document mentions multiple design
methods and criteria in use, factors, which may influence the
choice, are presented.
1.3—Scope
This document is limited in scope to the engineering,
construction, repair, and upgrade of dynamic equipment
foundations. For the purposes of this document, dynamic
equipment includes the following:

1. Rotating machinery;
2. Reciprocating machinery; and
3. Impact or impulsive machinery.
1.4—Notation
[C] = damping matrix
[K] = stiffness matrix
[K
*
] = impedance with respect to CG
[k] = reduced stiffness matrix
[k
j
′ ] = battered pile stiffness matrix
[M] = mass matrix
[m] = reduced mass matrix
[T] = transformation matrix for battered pile
[α
ir
] = matrix of interaction factors between any
two piles with diagonal terms α
ii
= 1
A = displacement amplitude
A
head
, A
crank
= head and crank areas, in.
2
(mm

2
)
A
p
= cross-sectional area of the pile
a, b = plan dimensions of a rectangular foundation
a
o
= dimensionless frequency
B
c
= cylinder bore diameter, in. (mm)
B
i
= mass ratio for the i-th direction
B
r
= ram weight, tons (kN)
b
1,
b
2
= 0.425 and 0.687, Eq. (4.15d)
c
gi
= damping of pile group in the i-th direction
c
i
= damping constant for the i-th direction
c

i
*
(adj) = damping in the i-th direction adjusted for
material damping
c
ij
= equivalent viscous damping of pile j in the
i-th direction
D
i
= damping ratio for the i-th direction
D
rod
= rod diameter, in. (mm)
d = pile diameter
d
n
= nominal bolt diameter, in. (m)
d
s
= displacement of the slide, in. (mm)
E
p
= Young’s modulus of the pile
e
m
= mass eccentricity, in. (mm)
e
v
= void ratio

F = time varying force vector
F
1
= correction factor
F
block
= the force acting outwards on the block from
which concrete stresses should be calcu-
lated, lbf (N)
(F
bolt
)
CHG
= the force to be restrained by friction at the
cross head guide tie-down bolts, lbf (N)
(F
bolt
)
frame
= the force to be restrained by friction at the
frame tie-down bolts, lbf (N)
F
D
= damper force
F
GMAX
= maximum horizontal gas force on a throw
or cylinder, lbf (N)
F
IMAX

= maximum horizontal inertia force on a
throw or cylinder, lbf (N)
F
o
= dynamic force amplitude (zero-to-peak),
lbf (N)
F
r
= maximum horizontal dynamic force
F
red
= a force reduction factor with suggested
value of 2, to account for the fraction of
individual cylinder load carried by the
compressor frame (“frame rigidity
factor”)
F
rod
= force acting on piston rod, lbf (N)
F
s
= dynamic inertia force of slide, lbf (N)
F
THROW
= horizontal force to be resisted by each
throw’s anchor bolts, lbf (N)
F
unbalance
= the maximum value from Eq. (3.18)
applied using parameters for a horizontal

compressor cylinder, lbf (N)
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-3
f
i1
, f
i2
= dimensionless stiffness and damping
functions for the i-th direction, piles
f
m
= frequency of motion, Hz
f
n
= system natural frequency (cycles per second)
f
o
= operating speed, rpm
G = dynamic shear modulus of the soil
G
ave
= the average value of shear modulus of the
soil over the pile length
G
c
= the average value of shear modulus of the
soil over the critical length
GE = pile group efficiency
G
l
= soil shear modulus at tip of pile

G
p
J=torsional stiffness of the pile
G
s
= dynamic shear modulus of the embedment
(side) material
G
z
= the shear modulus at depth z = l
c
/4
H = depth of soil layer
I
i
= mass moment of inertia of the machine-
foundation system for the i-th direction
I
p
= moment of inertia of the pile cross section
i =
i=a directional indicator or modal indicator,
Eq. (4.48), as a subscript
K
2
= a parameter that depends on void ratio and
strain amplitude
K
eff
= the effective bearing stiffness, lbf/in. (N/mm)

K
ij
*
= impedance in the i-th direction with respect
to motion of the CG in j-th direction
K
n
= nut factor for bolt torque
K
uu
= horizontal spring constant
K
uψ
= coupling spring constant
K
ψψ
= rocking spring constant
k = the dynamic stiffness provided by the
supporting media
k
ei
*
= impedance in the i-th direction due to
embedment
k
gi
= pile group stiffness in the i-th direction
k
i
= stiffness for the i-th direction

k
i
(adj) = stiffness in the i-th direction adjusted for
material damping
k
i
*
= complex impedance for the i-th direction
k
i
*
(adj) = impedance adjusted for material damping
k
ij
= stiffness of pile j in the i-th direction
k
j
= battered pile stiffness matrix
k
r
= stiffness of individual pile considered in
isolation
k
st
= static stiffness constant
k
vj
= vertical stiffness of a single pile
L = length of connecting rod, in. (mm)
L

B
= the greater plan dimension of the founda-
tion block, ft (m)
L
i
= length of the connecting rod of the crank
mechanism at the i-th cylinder
l = depth of embedment (effective)
l
c
= critical length of a pile
l
p
= pile length
M
h
= hammer mass including any auxiliary
foundation, lbm (kg)
M
r
= ram mass including dies and ancillary
parts, lbm (kg)
m = mass of the machine-foundation system
m
d
= slide mass including the effects of any
balance mechanism, lbm (kg)
m
r
= rotating mass, lbm (kg)

m
rec,i
= reciprocating mass for the i-th cylinder
m
rot,i
= rotating mass of the i-th cylinder
m
s
= effective mass of a spring
(N
bolt
)
CHG
= the number of bolts holding down one
crosshead guide
(N
bolt
)
frame
= the number of bolts holding down the
frame, per cylinder
NT = normal torque, ft-lbf (m-N)
P
head
, P
crank
= instantaneous head and crank pressures,
psi (µPa)
P
s

= power being transmitted by the shaft at the
connection, horsepower (kilowatts)
R, R
i
= equivalent foundation radius
r = length of crank, in. (mm)
r
i
= radius of the crank mechanism of the i-th
cylinder
r
o
= pile radius or equivalent radius
S = press stroke, in. (mm)
S
f
= service factor, used to account for increasing
unbalance during the service life of the
machine, generally greater than or equal to 2
S
i1
, S
i2
= dimensionless parameters (Table 4.2)
s = distance between piles
T = foundation thickness, ft (m)
T
b
= bolt torque, lbf-in. (N-m)
T

min
= minimum required anchor bolt tension
t =time, s
V
max
= the maximum allowable vibration, in. (mm)
V
s
= shear wave velocity of the soil, ft/s (m/s)
v = displacement amplitude
v′ = velocity, in./s (cm/s)
v
h
= post-impact hammer velocity, in./s (mm/s)
v
o
= reference velocity = 18.4 ft/s (5.6 m/s)
from a free fall of 5.25 ft (1.6 m)
v
r
= ram impact velocity, ft/s (m/s)
W = strain energy
W
a
= equipment weight at anchorage location
W
f
= weight of the foundation, tons (kN)
W
p

= bolt preload, lbf (N)
W
r
= rotating weight, lbf (N)
w = soil weight density
X = vector representation of time-dependent
displacements for MDOF systems
X
i
= distance along the crankshaft from the
reference origin to the i-th cylinder
x, z = the pile coordinates indicated in Fig. 4.9
x
r
, z
r
= pile location reference distances
y
c
= distance from the CG to the base support
y
e
= distance from the CG to the level of
embedment resistance
y
p
= crank pin displacement in local Y-axis,
in. (mm)
1–
351.3R-4 ACI COMMITTEE REPORT

Z
p
= piston displacement, in. (mm)
z
p
= crank pin displacement in local Z-axis, in.
(mm)
α = the angle between a battered pile and
vertical
α′ = modified pile group interaction factor
α
1
= coefficient dependent on Poisson’s ratio
as given in Table 4.1
α
h
= ram rebound velocity relative to impact
velocity
α
i
= the phase angle for the crank radius of the
i-th cylinder, rad
α
ij
*
= complex pile group interaction factor for
the i-th

pile to the j-th pile
α

uf
= the horizontal interaction factor for fixed-
headed piles (no head rotation)
α
uH
=the horizontal interaction factor due to
horizontal force (rotation allowed)
α
v
= vertical interaction coefficient between
two piles
α
ψH
=the rotation due to horizontal force
α
ψM
= the rotation due to moment
β = system damping ratio
β
i
= rectangular footing coefficients (Richart,
Hall, and Woods 1970), i = v, u, or ψ
β
j
= coefficient dependent on Poisson’s ratio
as given in Table 4.1, j = 1 to 4
β
m
= material damping ratio of the soil
β

p
= angle between the direction of the loading
and the line connecting the pile centers
δ = loss angle
∆W = area enclosed by the hysteretic loop
ε
ir
= the elements of the inverted matrix [α
ir
]
–1
ψ
i
= reduced mode shape vector for the i-th
mode
γ
j
= coefficient dependent on Poisson’s ratio
as given in Table 4.1, j = 1 to 4
λ = pile-soil stiffness ratio (E
p
/G
l
)
µ = coefficient of friction
ν = Poisson’s ratio of the soil
ν
s
= Poisson’s ratio of the embedment (side)
material

ρ = soil mass density (soil weight density/gravi-
tational acceleration)
ρ
a
= G
ave
/G
l
ρ
c
=G
z
/G
c
σ
o
= probable confining pressure, lbf/ft
2
(Pa)
ω
i
= circular natural frequency for the i-th
mode
ω
m
= circular frequency of motion
ω
n
= circular natural frequencies of the system
ω

o
= circular operating frequency of the
machine (rad/s)
ω
su
, ω
sv
= circular natural frequencies of a soil layer
in u and v directions
CHAPTER 2—FOUNDATION AND MACHINE TYPES
2.1—General considerations
The type, configuration, and installation of a foundation or
support structure for dynamic machinery may depend on the
following factors:
1. Site conditions such as soil characteristics, topography,
seismicity, climate, and other effects;
2. Machine base configuration such as frame size,
cylinder supports, pulsation bottles, drive mechanisms,
and exhaust ducts;
3. Process requirements such as elevation requirements
with respect to connected process equipment and hold-down
requirements for piping;
4. Anticipated loads such as the equipment static weight,
and loads developed during erection, startup, operation,
shutdown, and maintenance;
5. Erection requirements such as limitations or constraints
imposed by construction equipment, procedures, techniques,
or the sequence of erection;
6. Operational requirements such as accessibility, settle-
ment limitations, temperature effects, and drainage;

7. Maintenance requirements such as temporary access,
laydown space, in-plant crane capabilities, and machine
removal considerations;
8. Regulatory factors or building code provisions such as
tied pile caps in seismic zones;
9. Economic factors such as capital cost, useful or antici-
pated life, and replacement or repair cost;
10. Environmental requirements such as secondary
containment or special concrete coating requirements; and
11. Recognition that certain machines, particularly large
reciprocating compressors, rely on the foundation to add
strength and stiffness that is not inherent in the structure of
the machine.
2.2—Machine types
2.2.1 Rotating machinery—This category includes gas
turbines, steam turbines, and other expanders; turbo-pumps
and compressors; fans; motors; and centrifuges. These
machines are characterized by the rotating motion of impel-
lers or rotors.
Unbalanced forces in rotating machines are created when
the mass centroid of the rotating part does not coincide with
the center of rotation (Fig. 2.1). This dynamic force is a function
of the shaft mass, speed of rotation, and the magnitude of the
offset. The offset should be minor under manufactured
conditions when the machine is well balanced, clean, and
without wear or erosion. Changes in alignment, operation
near resonance, blade loss, and other malfunctions or
undesirable conditions can greatly increase the force applied
to its bearings by the rotor. Because rotating machines
normally trip and shut down at some vibration limit, a real-

istic continuous dynamic load on the foundation is that
resulting from vibration just below the trip level.
2.2.2 Reciprocating machinery—For reciprocating
machinery, such as compressors and diesel engines, a piston
moving in a cylinder interacts with a fluid through the
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-5
kinematics of a slider crank mechanism driven by, or
driving, a rotating crankshaft.
Individual inertia forces from each cylinder and each
throw are inherently unbalanced with dominant frequencies
at one and two times the rotational frequency (Fig. 2.2).
Reciprocating machines with more than one piston require
a particular crank arrangement to minimize unbalanced
forces and moments. A mechanical design that satisfies
operating requirements should govern. This leads to piston/
cylinder assemblies and crank arrangements that do not
completely counter-oppose; therefore, unbalanced loads
occur, which should be resisted by the foundation.
Individual cylinder fluid forces act outward on the
cylinder head and inward on the crankshaft (Fig. 2.2). For a
rigid cylinder and frame these forces internally balance, but
deformations of large machines can cause a significant
portion of the fluid load to be transmitted to the mounts and
into the foundation. Particularly on large reciprocating
compressors with horizontal cylinders, it is inappropriate
and unconservative to assume the compressor frame and
cylinder are sufficiently stiff to internally balance all forces.
Such an assumption has led to many inadequate mounts for
reciprocating machines.
2.2.3 Impulsive machinery—Equipment, such as forging

hammers and some metal-forming presses, operate with
regulated impacts or shocks between different parts of the
equipment. This shock loading is often transmitted to the
foundation system of the equipment and is a factor in the
design of the foundation.
Closed die forging hammers typically operate by dropping
a weight (ram) onto hot metal, forcing it into a predefined
shape. While the intent is to use this impact energy to form
and shape the material, there is significant energy transmission,
particularly late in the forming process. During these final
blows, the material being forged is cooling and less shaping
takes place. Thus, pre-impact kinetic energy of the ram
converts to post-impact kinetic energy of the entire forging
hammer. As the entire hammer moves downward, it
becomes a simple dynamic mass oscillating on its supporting
medium. This system should be well damped so that the
oscillations decay sufficiently before the next blow. Timing
of the blows commonly range from 40 to 100 blows per min.
The ram weights vary from a few hundred pounds to 35,000 lb
(156 kN). Impact velocities in the range of 25 ft/s (7.6 m/s)
are common. Open die hammers operate in a similar fashion
but are often of two-piece construction with a separate
hammer frame and anvil.
Forging presses perform a similar manufacturing function
as forging hammers but are commonly mechanically or
hydraulically driven. These presses form the material at low
velocities but with greater forces. The mechanical drive
system generates horizontal dynamic forces that the engineer
should consider in the design of the support system. Rocking
stability of this construction is important. Figure 2.3 shows a

typical horizontal forcing function through one full stroke of
a forging press.
Mechanical metal forming presses operate by squeezing
and shearing metal between two dies. Because this equip-
ment can vary greatly in size, weight, speed, and operation,
the engineer should consider the appropriate type. Speeds
can vary from 30 to 1800 strokes per min. Dynamic forces from
the press develop from two sources: the mechanical balance of
the moving parts in the equipment and the response of the
press frame as the material is sheared (snap-through forces).
Imbalances in the mechanics of the equipment can occur both
horizontally and vertically. Generally high-speed equipment
is well balanced. Low-speed equipment is often not balanced
because the inertia forces at low speeds are small. The
dynamic forces generated by all of these presses can be
significant as they are transmitted into the foundation and
propagated from there.
2.2.4 Other machine types—Other machinery generating
dynamic loads include rock crushers and metal shredders.
While part of the dynamic load from these types of equipment
tend to be based on rotating imbalances, there is also a
Fig. 2.1—Rotating machine diagram.
Fig. 2.2—Reciprocating machine diagram.
Fig. 2.3—Forcing function for a forging press.
351.3R-6 ACI COMMITTEE REPORT
random character to the dynamic signal that varies with the
particular operation.
2.3—Foundation types
2.3.1 Block-type foundation (Fig. 2.4)—Dynamic machines
are preferably located close to grade to minimize the elevation

difference between the machine dynamic forces and the center
of gravity of the machine-foundation system. The ability to use
such a foundation primarily depends on the quality of near
surface soils. Block foundations are nearly always designed as
rigid structures. The dynamic response of a rigid block
foundation depends only on the dynamic load, foundation’s
mass, dimensions, and soil characteristics.
2.3.2 Combined block-type foundation (Fig. 2.5)—
Combined blocks are used to support closely spaced
machines. Combined blocks are more difficult to design
because of the combination of forces from two or more
machines and because of a possible lack of stiffness of a
larger foundation mat.
2.3.3 Tabletop-type foundation (Fig. 2.6)—Elevated
support is common for large turbine-driven equipment such
as electric generators. Elevation allows for ducts, piping, and
ancillary items to be located below the equipment. Tabletop
structures are considered to be flexible, hence their response
to dynamic loads can be quite complex and depend both on
the motion of its discreet elements (columns, beams, and
footing) and the soil upon which it is supported.
2.3.4 Tabletop with isolators (Fig. 2.7)—Isolators (springs
and dampers) located at the top of supporting columns are
sometimes used to minimize the response to dynamic loading.
The effectiveness of isolators depends on the machine speed
and the natural frequency of the foundation. Details of this
type of support are provided in Section 4.5.
2.3.5 Spring-mounted equipment (Fig. 2.8)—Occasionally
pumps are mounted on springs to minimize thermal forces
from connecting piping. The springs are then supported on a

block-type foundation. This arrangement has a dynamic
effect similar to that for tabletops with vibration isolators.
Other types of equipment are spring mounted to limit the
transmission of dynamic forces.
2.3.6 Inertia block in structure (Fig. 2.9)—Dynamic equip-
ment on a structure may be relatively small in comparison to the
overall size of the structure. In this situation, dynamic machines
are usually designed with a supporting inertia block to alter
natural frequencies away from machine operating speeds and
resist amplitudes by increasing the resisting inertia force.
2.3.7 Pile foundations (Fig. 2.10)—Any of the previously
mentioned foundation types may be supported directly on soil
or on piles. Piles are generally used where soft ground condi-
Fig. 2.4—Block-type foundation.
Fig. 2.5—Combined block foundation.
Fig. 2.6—Tabletop foundation.
Fig. 2.7—Tabletop with isolators.
Fig. 2.8—Spring-mounted block formation.
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-7
tions result in low allowable contact pressures and excessive
settlement for a mat-type foundation. Piles use end bearing,
frictional side adhesion, or a combination of both to transfer
axial loads into the underlying soil. Transverse loads are
resisted by soil pressure bearing against the side of the pile
cap or against the side of the piles. Various types of piles are
used including drilled piers, auger cast piles, and driven piles.
CHAPTER 3—DESIGN CRITERIA
3.1—Overview of design criteria
The main issues in the design of concrete foundations that
support machinery are defining the anticipated loads, estab-

lishing the performance criteria, and providing for these
through proper proportioning and detailing of structural
members. Yet, behind this straightforward definition lies the
need for careful attention to the interfaces between machine,
mounting system, and concrete foundation.
The loads on machine foundations may be both static and
dynamic. Static loads are principally a function of the
weights of the machine and all its auxiliary equipment.
Dynamic loads, which occur during the operation of the
machine, result from forces generated by unbalance, inertia
of moving parts, or both, and by the flow of fluid and gases
for some machines. The magnitude of these dynamic loads
primarily depends upon the machine’s operating speed and
the type, size, weight, and arrangement (position) of moving
parts within the casing.
The basic goal in the design of a machine foundation is to
limit its motion to amplitudes that neither endanger the satis-
factory operation of the machine nor disturb people working in
the immediate vicinity (Gazetas 1983). Allowable amplitudes
depend on the speed, location, and criticality or function of
the machine. Other limiting dynamic criteria affecting the
design may include avoiding resonance and excessive trans-
missibility to the supporting soil or structure. Thus, a key
ingredient to a successful design is the careful engineering
analysis of the soil-foundation response to dynamic loads
from the machine operation.
The foundation’s response to dynamic loads can be signif-
icantly influenced by the soil on which it is constructed.
Consequently, critical soil parameters, such as the dynamic
soil shear modulus, are preferably determined from a field

investigation and laboratory tests rather than relying on
generalized correlations based on broad soil classifications.
Due to the inherent variability of soil, the dynamic response
of machine foundations is often evaluated using a range of
values for the critical soil properties.
Furthermore, a machinery support structure or foundation is
designed with adequate structural strength to resist the worst
possible combination of loads occurring over its service life.
This often includes limiting soil-bearing pressures to well
within allowable limits to ensure a more predictable dynamic
response and prevent excessive settlements and soil failures.
Additionally, concrete members are designed and detailed to
prevent cracking due to fatigue and stress reversals caused by
dynamic loads, and the machine’s mounting system is designed
and detailed to transmit loads from the machine into the
foundation, according to the criteria in Section 3.6.
3.2—Foundation and equipment loads
Foundations supporting reciprocating or rotating compressors,
turbines, generators and motors, presses, and other machinery
should withstand all the forces that may be imposed on them
during their service life. Machine foundations are unique
because they may be subjected to significant dynamic loads
during operation in addition to normal design loads of
gravity, wind, and earthquake. The magnitude and charac-
teristics of the operating loads depend on the type, size,
speed, and layout of the machine.
Generally, the weight of the machine, center of gravity,
surface areas, and operating speeds are readily available
from the manufacturer of the machine. Establishing appro-
priate values for dynamic loads is best accomplished through

careful communication and clear understanding between the
machine manufacturer and foundation design engineer as to
the purpose, and planned use for the requested information,
and the definition of the information provided. It is in the best
interests of all parties (machine manufacturer, foundation
design engineer, installer, and operator) to ensure effective
definition and communication of data and its appropriate use.
Machines always experience some level of unbalance, vibra-
tion, and force transmitted through the bearings. Under some
off-design conditions, such as wear, the forces may increase
significantly. The machine manufacturer and foundation
design engineer should work together so that their combined
knowledge achieves an integrated system structure which
robustly serves the needs of its owner and operator and with-
stands all expected loads.
Sections 3.2.1 to 3.2.6 provide commonly used methods
for determining machine-induced forces and other design
Fig. 2.9—Inertia block in structure.
Fig. 2.10—Pile-supported foundation.
351.3R-8 ACI COMMITTEE REPORT
loads for foundations supporting machinery. They include
definitions and other information on dynamic loads to be
requested from the machine manufacturer and alternative
assumptions to apply when such data are unavailable or are
under-predicted.
3.2.1 Static loads
3.2.1.1 Dead loads—A major function of the foundation
is to support gravity (dead) loads due to the weight of the
machine, auxiliary equipment, pipe, valves, and deadweight
of the foundation structure. The weights of the machine compo-

nents are normally supplied by the machine manufacturer. The
distribution of the weight of the machine on the foundation
depends on the location of support points (chocks, soleplates)
and on the flexibility of the machine frame. Typically, there
are multiple support points, and, thus, the distribution is
statically indeterminate. In many cases, the machine manufac-
turer provides a loading diagram showing the vertical loads at
each support point. When this information is not available, it is
common to assume the machine frame is rigid and that its
weight is appropriately distributed between support points.
3.2.1.2 Live loads—Live loads are produced by personnel,
tools, and maintenance equipment and materials. The live loads
used in design should be the maximum loads expected during
the service life of the machine. For most designs, live loads are
uniformly distributed over the floor areas of platforms of
elevated support structures or to the access areas around at-
grade foundations. Typical live loads vary from 60 lbf/ft
2
(2.9 kPa) for personnel to as much as 150 lbf/ft
2
(7.2 kPa) for
maintenance equipment and materials.
3.2.1.3 Wind loads—Loads due to wind on the surface
areas of the machine, auxiliary equipment, and the support
foundation are based on the design wind speed for the partic-
ular site and are normally calculated in accordance with the
governing local code or standard. Wind loads rarely govern
the design of machine foundations except, perhaps, when the
machine is located in an enclosure that is also supported by
the foundation.

When designing machine foundations and support structures,
most practitioners use the wind load provisions of ASCE 7. The
analytical procedure of ASCE 7 provides wind pressures and
forces for use in the design of the main wind-force resisting
systems and anchorage of machine components.
Most structural systems involving machines and machine
foundations are relatively stiff (natural frequency in the
lateral direction greater than 1 Hz). Consequently, the
systems can be treated as rigid with respect to the wind gust
effect factor, and simplified procedures can be used. If the
machine is supported on flexible isolators and is exposed to
the wind, the rigid assumption may not be reasonable, and
more elaborate treatment of the gust effects is necessary as
described in ASCE 7 for flexible structural systems.
Appropriate consideration of the exposure conditions and
importance factors is also required to be consistent with the
facilities requirements.
3.2.1.4 Seismic loads—Machinery foundations located
in seismically active regions are analyzed for seismic loads.
Before 2000, these loads were determined in accordance
with methods prescribed in one of various regional building
codes (such as the UBC, the SBC, or the NBC) and standards
such as ASCE 7 and SEAOC Blue Book.
The publication of the IBC 2000 provides building officials
with the opportunity to replace the former regional codes
with a code that has nationwide applicability. The seismic
requirements in IBC 2000 and ASCE 7-98 are essentially
identical, as both are based on the 1997 NEHRP (FEMA 302)
provisions.
The IBC and its reference documents contain provisions

for design of nonstructural components, including dynamic
machinery, for seismic loads. For machinery supported
above grade or on more flexible elevated pedestals, seismic
amplification factors are also specified.
3.2.1.5 Static operating loads—Static operating loads
include the weight of gas or liquid in the machinery equipment
during normal operation and forces, such as the drive torque
developed by some machines at the connection between the
drive mechanism and driven machinery. Static operating
loads can also include forces caused by thermal growth of
the machinery equipment and connecting piping. Time-
varying (dynamic) loads generated by machines during
operation are covered elsewhere in this report.
Machines such as compressors and generators require
some form of drive mechanism, either integral with the
machine or separate from it. When the drive mechanism is
nonintegral, such as a separate electric motor, reciprocating
engine, and gas or steam turbine, it produces a net external
drive torque on the driven machine. The torque is equal in
magnitude and opposite in direction on the driver and driven
machine. The normal torque (sometimes called drive torque)
is generally applied to the foundation as a static force couple
in the vertical direction acting about the centerline of the
shaft of the machine. The magnitude of the normal torque is
often computed from the following formula
NT = lbf-ft (3-1)
NT = N-m
where
NT = normal torque, ft-lbf (m-N);
P

s
= power being transmitted by the shaft at the
connection, horsepower (kilowatts); and
f
o
= operating speed, rpm.
The torque load is generally resolved into a vertical force
couple by dividing it by the center-to-center distance between
longitudinal soleplates or anchor points (Fig. 3.1(a)). When the
machine is supported by transverse soleplates only, the
torque is applied along the width of the soleplate assuming a
straight line variation of force (Fig. 3.1(b)). Normal torque
can also be caused by jet forces on turbine blades. In this case
it is applied to the foundation in the opposite direction from
the rotation of the rotor.
The torque on a generator stator is applied in the same
direction as the rotation of the rotor and can be high due to
5250()P
s
()
f
o

9550()P
s
()
f
o

FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-9

startup or an electrical short circuit. Startup torque, a property
of electric motors, should be obtained from the motor
manufacturer. The torque created by an electrical short
circuit is considered a malfunction, emergency, or accidental
load and is generally reported separately by the machinery
manufacturer. Often in the design for this phenomenon, the
magnitude of the emergency drive torque is determined by
applying a magnification factor to the normal torque.
Consultation with the generator manufacturer is necessary to
establish the appropriate magnification factor.
3.2.1.6 Special loads for elevated-type foundations—To
ensure adequate strength and deflection control, the
following special static loading conditions are recommended
in some proprietary standards for large equipment on
elevated-type foundations:
1. Vertical force equal to 50% of the total weight of each
machine;
2. Horizontal force (in the transverse direction) equal to
25% of the total weight of each machine; and
3. Horizontal force (in the longitudinal direction) equal to
25% of the total weight of each machine.
These forces are additive to normal gravity loads and are
considered to act at the centerline of the machine shaft.
Loads 1, 2, and 3 are not considered to act concurrently with
one another.
3.2.1.7 Erection and maintenance loads—Erection and
maintenance loads are temporary loads from equipment,
such as cranes and forklifts, required for installing or
dismantling machine components during erection or mainte-
nance. Erection loads are usually furnished in the manufac-

turer’s foundation load drawing and should be used in
conjunction with other specified dead, live, and environmental
loads. Maintenance loads occur any time the equipment is
being drained, cleaned, repaired, and realigned or when the
components are being removed or replaced. Loads may
result from maintenance equipment, davits, and hoists. Envi-
ronmental loads, such as full wind and earthquake, are not
usually assumed to act with maintenance loads, which gener-
ally occur for only a relatively short duration.
3.2.1.8 Thermal loads—Changing temperatures of
machines and their foundations cause expansions and
contractions, and distortions, causing the various parts to try
to slide on the support surfaces. The magnitude of the
resulting frictional forces depends on the magnitude of the
temperature change, the location of the supports, and on the
condition of the support surfaces. The thermal forces do not
impose a net force on the foundation to be resisted by soil or
piles because the forces on any surface are balanced by equal
and opposite forces on other support surfaces. Thermal
forces, however, may govern the design of the grout system,
pedestals, and hold downs.
Calculation of the exact thermal loading is very difficult
because it depends on a number of factors, including
distance between anchor points, magnitude of temperature
change, the material and condition of the sliding surface, and
the magnitude of the vertical load on each soleplate. Lacking
a rigorous analysis, the magnitude of the frictional load may
be calculated as follows
Force = (friction coefficient)(load acting through soleplate) (3-2)
The friction coefficient generally varies from 0.2 to 0.5.

Loads acting through the soleplate include: machine dead
load, normal torque load, anchor bolt load, and piping loads.
Heat transfer to the foundation can be by convection across an
air gap (for example, gap between sump and block) and by
conduction through points of physical contact. The resultant
temperature gradients induce deformations, strains, and stresses.
When evaluating thermal stress, the calculations are
strongly influenced by the stiffness and restraint against
deformation for the structural member in question. There-
fore, it is important to consider the self-relieving nature of
thermal stress due to deformation to prevent being overly
conservative in the analysis. As the thermal forces are
applied to the foundation member by the machine, the foun-
dation member changes length and thereby provides reduced
resistance to the machine forces. This phenomenon can have
the effect of reducing the thermal forces from the machine.
Accurate determinations of concrete surface temperatures
and thermal gradients are also important. Under steady-state
normal operating conditions, temperature distributions
across structural sections are usually linear. The air gap
between the machine casing and foundation provides a
significant means for dissipating heat, and its effect should
be included when establishing surface temperatures.
Normally, the expected thermal deflection at various
bearings is estimated by the manufacturer, based on past
field measurements on existing units. The machine erector
then compensates for the thermal deflection during installation.
Fig. 3.1—Equivalent forces for torque loads.
351.3R-10 ACI COMMITTEE REPORT
Reports are available (Mandke and Smalley 1992;

Mandke and Smalley 1989; and Smalley 1985) that illustrate
the effects of thermal loads and deflections in the concrete
foundation of a large reciprocating compressor and their
influence on the machine.
3.2.2 Rotating machine loads—Typical heavy rotating
machinery include centrifugal air and gas compressors, hori-
zontal and vertical fluid pumps, generators, rotating steam and
gas turbine drivers, centrifuges, electric motor drivers, fans,
and blowers. These types of machinery are characterized by
the rotating motion of one or more impellers or rotors.
3.2.2.1 Dynamic loads due to unbalanced masses—
Unbalanced forces in rotating machines are created when the
mass centroid of the rotating part does not coincide with the
axis of rotation. In theory, it is possible to precisely balance
the rotating elements of rotating machinery. In practice, this
is never achieved; slight mass eccentricities always remain.
During operation, the eccentric rotating mass produces
centrifugal forces that are proportional to the square of
machine speed. Centrifugal forces generally increase during
the service life of the machine due to conditions such as
machine wear, rotor play, and dirt accumulation.
A rotating machine transmits dynamic force to the foundation
predominantly through its bearings (with small, generally
unimportant exceptions such as seals and the air gap in a
motor). The forces acting at the bearings are a function of the
level and axial distribution of unbalance, the geometry of the
rotor and its bearings, the speed of rotation, and the detailed
dynamic characteristics of the rotor-bearing system. At or near
a critical speed, the force from rotating unbalance can be
substantially amplified, sometimes by a factor of five or more.

Ideally, the determination of the transmitted force under
different conditions of unbalance and at different speeds
results from a dynamic analysis of the rotor-bearing system,
using an appropriate combination of computer programs for
calculating bearing dynamic characteristics and the response
to unbalance of a flexible rotor in its bearings. Such an analysis
would usually be performed by the machine manufacturer.
Results of such analyses, especially values for transmitted
bearing forces, represent the best source of information for
use by the foundation design engineer. This and other
approaches used in practice to quantify the magnitude of
dynamic force transmitted to the foundation are discussed in
Sections 3.2.2.1a to 3.2.2.1.3e.
3.2.2.1a Dynamic load provided by the manufac-
turer—The engineer should request and the machine manu-
facturer should provide the following information:
Design levels of unbalance and basis—This information
documents the unbalance level the subsequent transmitted
forces are based on.
Dynamic forces transmitted to the bearing pedestals
under the following conditions—
a) Under design unbalance levels over operating speed
range;
b) At highest vibration when negotiating critical speeds;
c) At a vibration level where the machine is just short of
tripping on high vibration; and
d) Under the maximum level of upset condition the
machine is designed to survive (for example, loss of one or
more blades).
Items a and b document the predicted dynamic forces

resulting from levels of unbalance assumed in design for
normal operation. Using these forces, it is possible to predict
the normal dynamic vibration of the machine on its foundation.
Item c identifies a maximum level of transmitted force
with which the machine could operate continuously without
tripping; the foundation should have the strength to tolerate
such a dynamic force on a continuous basis.
Item d identifies the higher level of dynamic force, which
could occur under occasional upset conditions over a short
period of time. If the machine is designed to tolerate this
level of dynamic force for a short period of time, then the
foundation should also be able to tolerate it for a similar
period of time.
If an independent dynamic analysis of the rotor-bearing
system is performed by the end user or by a third party, such
an analysis can provide some or all of the above dynamic
forces transmitted to the foundation.
By assuming that the dynamic force transmitted to the
bearings equals the rotating unbalanced force generated by
the rotor, information on unbalance can provide an estimate
of the transmitted force.
3.2.2.1b Machine unbalance provided by the manufac-
turer—When the mass unbalance (eccentricity) is known or
stated by the manufacturer, the resulting dynamic force
amplitude is
F
o
= m
r
e

m
ω
o
2
S
f
/12 lbf (3-3)
F
o
= m
r
e
m
ω
o
2
S
f
/1000 N
where
F
o
= dynamic force amplitude (zero-to-peak), lbf (N);
m
r
= rotating mass, lbm (kg);
e
m
= mass eccentricity, in. (mm);
ω

o
= circular operating frequency of the machine (rad/s);
and
S
f
= service factor, used to account for increased
unbalance during the service life of the machine,
generally greater than or equal to 2.
3.2.2.1c Machine unbalance meeting industry
criteria—Many rotating machines are balanced to an initial
balance quality either in accordance with the manufacturer’s
procedures or as specified by the purchaser. ISO 1940 and
ASA/ANSI S2.19 define balance quality in terms of a constant
e
m
ω
o
. For example, the normal balance quality Q for parts of
process-plant machinery is 0.25 in./s (6.3 mm/s). Other typical
balance quality grade examples are shown in Table 3.1. To
meet these criteria a rotor intended for faster speeds should be
better balanced than one operating at a slower speed. Using
this approach, Eq. (3-3) can be rewritten as
F
o
= m
r
Qω
o
S

f
/12 lbf (3-4)
F
o
= m
r
Qω
o
S
f
/1000 N
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-11
API 617 and API 684 work with maximum residual unbal-
ance U
max
criteria for petroleum processing applications. The
mass eccentricity is determined by dividing U
max
by the rotor
weight. For axial and centrifugal compressors with maximum
continuous operating speeds greater than 25,000 rpm, API 617
establishes a maximum allowable mass eccentricity of 10 ×
10
–6
in. (250 nm). For compressors operating at slower speeds,
the maximum allowable mass eccentricity is
e
m
= 0.25/f
o

in. (3-5)
e
m
= 6.35/f
o
mm
where
f
o
= operating speed, rpm ≤ 25,000 rpm.
This permitted initial mass eccentricity is tighter than ISO
balance quality grade G2.5, which would be applied to this
type of equipment (Table 3.1, turbo-compressors) under ISO
1940. As such, the dynamic force computed from this API
consideration will be quite small and a larger service factor
might be used to have a realistic design force.
API 617 also identifies a limitation on the peak-to-peak
vibration amplitude during mechanical testing of the
compressor with the equipment operating at its maximum
continuous speed ((12,000/f
o
)
0.5
in. [25.4(12,000/f
o
)
0.5
mm]).
Some design firms use this criterion and a service factor S
f

of 2.0 to compute the dynamic force amplitude as
(3-6)
where W
r
= rotating weight, lbf (N).
3.2.2.1d Dynamic load determined from an empirical
formula—Rotating machine manufacturers often do not
report the unbalance that remains after balancing. Conse-
quently, empirical formulas are frequently used to ensure
that foundations are designed for some minimum unbalance,
which generally includes some allowance for increasing
unbalance over time. One general purpose empirical method
assumes that balancing improves with machine speed and
that there is a linear relationship between the unbalanced
forces and the machine speed. The zero-to-peak centrifugal
force amplitude from one such commonly used expression is
(3-7)
Equations (3-3), (3-4), (3-6), and (3-7) appear to be very
different: the exponents on the speed of rotation vary from 1
to 1.5 to 2, constants vary widely, and different variables
appear. Some equations use mass, others use weight. In
reality, the equations are more similar than they appear.
Given the right understanding of Q as a replacement for eω,
Eq. (3-3), (3-4), and (3-7) take on the same character. These
equations then indicate that the design force at operating
speed varies linearly with both the mass of the rotating body
and the operating rotational speed. Once that state is identified,
Eq. (3-3) can be adjusted to reflect the actual speed of rotation,
and the dynamic centrifugal force is seen to vary with the
square of the speed. Restating Eq. (3-6) and (3-7) in the form

of Eq. (3-3) allows for the development of an effective eccen-
tricity implied within these equations with the comparison
shown in Fig. 3.2. Equation (3-7) produces the same result as
Eq. (3-4) using Q = 0.25 in./s (6.3 mm/s), and S
f
= 2.5.
The centrifugal forces due to mass unbalance are considered
to act at the center of gravity of the rotating part and vary
harmonically at the speed of the machine in the two orthogonal
directions perpendicular to the shaft. The forces in the two
orthogonal directions are equal in magnitude and 90 degrees
out of phase and are transmitted to the foundation through the
F
o
W
r
f
o
1.5
322,000
=
F
o
W
r
f
o
6000
=
Table 3.1—Balance quality grades for selected

groups of representative rigid rotors (excerpted
from ANSI/ASA S2.19)
Balance
quality
guide
Product of eω,
in./s (mm/s) Rotor types—general examples
G1600 63 (1600)
Crankshaft/drives of rigidly mounted, large,
two-cycle engines
G630 2.5 (630)
Crankshaft/drives of rigidly mounted, large,
four-cycle engines
G250 10 (250)
Crankshaft/drives of rigidly mounted, fast,
four-cylinder diesel engines
G100 4 (100)
Crankshaft/drives of fast diesel engines with six or
more cylinders
G40 1.6 (40)
Crankshaft/drives of elastically mounted, fast
four-cycle engines (gasoline or diesel) with six or
more cylinders
G16 0.6 (16)
Parts of crushing machines; drive shafts (propeller
shafts, cardan shafts) with special requirements;
crankshaft/drives of engines with six or more
cylinders under special requirements
G6.3 0.25 (6.3)
Parts of process plant machines; centrifuge drums,

paper machinery rolls, print rolls; fans; flywheels;
pump impellers; machine tool and general
machinery parts; medium and large electric
armatures (of electric motors having at least
80 mm shaft height) without special requirement
G2.5 0.1 (2.5)
Gas and steam turbines, including marine main
turbines; rigid turbo-generator rotors; turbo-
compressors; machine tool drives; medium and
large electric armatures with special requirements;
turbine driven pumps
G1 0.04 (1) Grinding machine drives
G0.4 0.015 (0.4)
Spindles, discs, and armatures of precision
grinders
Fig. 3.2—Comparison of effective eccentricity.
351.3R-12 ACI COMMITTEE REPORT
bearings. Schenck (1990) provides useful information about
balance quality for various classes of machinery.
3.2.2.1e Machine unbalance determined from trip
vibration level and effective bearing stiffness—Because a
rotor is often set to trip on high vibration, it can be expected
to operate continuously at any vibration level up to the trip
limit. Given the effective bearing stiffness, it is possible to
calculate the maximum dynamic force amplitude as
F
o
= V
max
K

eff
(3-8)
where
V
max
= the maximum allowable vibration, in. (mm); and
K
eff
= the effective bearing stiffness, lbf/in. (N/mm).
To use this approach, the manufacturer should provide
effective bearing stiffness or the engineer should calculate it
from the bearing geometry and operating conditions (such as
viscosity and speed).
3.2.2.2 Loads from multiple rotating machines—If a
foundation supports multiple rotating machines, the engineer
should compute unbalanced force based on the mass, unbal-
ance, and operating speed of each rotating component. The
response to each rotating mass is then combined to determine
the total response. Some practitioners, depending on the
specific situation of machine size and criticality, find it
advantageous to combine the unbalanced forces from each
rotating component into a single resultant unbalanced force.
The method of combining two dynamic forces is up to indi-
vidual judgment and often involves some approximations. In
some cases, loads or responses can be added absolutely. In
other cases, the loads are treated as out-of-phase so that
twisting effects are increased. Often, the operating speed of
the equipment should be considered. Even if operating
speeds are nominally the same, the design engineer should
recognize that during normal operation, the speed of the

machines will vary and beating effects can develop. Beating
effects develop as two machines operate at close to the same
speed. At one point in time, responses to the two machines are
additive and motions are maximized. A short time later, the
responses cancel each other and the motions are minimized.
The net effect is a continual cyclic rising and falling of motion.
3.2.3 Reciprocating machine loads—Internal-combustion
engines, piston-type compressors and pumps, some metal
forming presses, steam engines, and other machinery are
characterized by the rotating motion of a master crankshaft
and the linear reciprocating motion of connected pistons or
sliders. The motion of these components cause cyclically
varying forces, often called reciprocating forces.
3.2.3.1 Primary and secondary reciprocating loads—
The simplest type of reciprocating machine uses a single
crank mechanism as shown in Fig. 3.3. The idealization of
this mechanism consists of a piston that moves within a
guiding cylinder, a crank of length r that rotates about a
crank shaft, and a connecting rod of length L. The connecting
rod is attached to the piston at point P and to the crank at
point C. The wrist pin P oscillates while the crank pin C
follows a circular path. This idealized single cylinder illus-
trates the concept of a machine producing both primary and
secondary reciprocating forces.
If the crank is assumed to rotate at a constant angular
velocity ω
o
, the translational acceleration of the piston along
its axis may be evaluated. If Z
p

is defined as the piston
displacement toward the crankshaft (local Z-axis), an
expression can be written for Z
p
at any time t. Further, the
velocity and acceleration can also be obtained by taking the first
and second derivatives of the displacement expression with
respect to time. The displacement, velocity, and acceleration
expressions for the motion of the piston are as follows
(3-9)
(3-10)

Z
p
(3-11)
where
Z
p
= piston displacement, in. (mm);
r = length of crank, in. (mm);
L = length of connecting rod, in. (mm);
ω
o
= circular operating frequency of the machine (rad/s); and
t = time, s.
Note that the expressions contain two terms each with a
sine or cosine; the term that varies with the frequency of the
rotation, ω
o
, is referred to as the primary term while the term

that varies at twice the frequency of rotation, 2ω
o
, is called
the secondary term.
Similar expressions can be developed for the local Z-axis
(parallel to piston movement) and local Y-axis (perpendic-
ular to piston movement) motion of the rotating parts of the
crank. If any unbalance in the crankshaft is replaced by a
mass concentrated at the crank pin C, such that the inertia
forces are the same as in the original system, the following
terms for motion at point C can be written
Z
p
r
r
2
4L
+


r ω
o
t
r
4L
2 ω
o
tcos+cos



–=
Z
·
p
rω
o
ω
o
t
r
2L
+2ω
o
tsinsin


=
rω
o
2
ω
o
t
r
L
2ω
o
tcoscos



=
Fig. 3.3—Crack mechanism.
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-13
y
p
= –r sinω
o
t (3-12)
= –r ωcosω
o
t (3-13)
= r ω
o
2
sinω
o
t (3-14)
z
p
= r(1 – cosω
o
t) (3-15)
= rω
o
sinω
o
t (3-16)
= r ω
o
2

cosω
o
t (3-17)
where
y
p
= crank pin displacement in local Y-axis, in. (mm); and
z
p
= crank pin displacement in local Z-axis, in. (mm).
Identifying a part of the connecting rod (usually 1/3 of its
mass) plus the piston as the reciprocating mass m
rec
concen-
trated at point P and designating the remainder of the
connecting rod plus the crank as the rotating mass m
rot
concentrated at point C, expressions for the unbalanced
forces are as follows
Parallel to piston movement
(3-18)
Perpendicular to piston movement
(3-19)
Note that Eq. (3.18) consists of two terms, a primary force
(m
rec
+ m
rot
)rω
o

2
cosω
o
t (3-20)
and a secondary force
(3-21)
whereas Eq. (3-19) has only a primary component.
3.2.3.2 Compressor gas loads—A reciprocating
compressor raises the pressure of a certain flow of gas by
imparting reciprocating motion on a piston within a cylinder.
The piston normally compresses gas during both directions
of reciprocating motion. As gas flows to and from each end,
the pressure of the gas increases as it is compressed by each
stroke of the piston. The increase in pressure within the
cylinder creates reaction forces on the head and crank ends
of the piston which alternate as gas flows to and from each
end of the cylinder.
y
p
·
y
p
··
z
p
·
z
p
··
F

z
m
rec
m
rot
+()rω
o
2
ω
o
tm
rec
r
2
ω
o
2
L
2ω
o
tcos+cos=
F
Y
m
rot
rω
o
2
ω
o

tsin=
m
rec
r
2
ω
o
2
L
2ω
o
tcos=
The gas force contributed to the piston rod equals the
instantaneous difference between the pressure force acting
on the head and crank end of the piston as shown in Fig. 3.4.
The following formulation can be used to estimate the
maximum force acting on the piston rod of an individual
double-acting cylinder
F
rod
= [(P
head
)(A
head
) – (P
crank
)(A
crank
)] F
1

(3-22)
A
head
= (π/4)B
c
2
(3-23)
A
crank
= (π/4)(B
c
2
– D
rod
2
) (3-24)
where
F
rod
= force acting on piston rod, lbf (N);
A
head
,
A
crank
= head and crank areas, in.
2
(mm
2
);

B
c
= cylinder bore diameter, in. (mm);
D
rod
= rod diameter, in. (mm);
P
head
,
P
crank
= instantaneous head and crank pressures, psi,
(MPa); and
F
1
= correction factor.
The head and crank end pressures vary continuously and
the differential force takes both positive and negative net
values during each cycle of piston motion. The normal
approach is to establish the head and crank pressures using
the maximum and minimum suction and discharge pressures.
For design purposes, it is common to multiply Eq. (3-22) by a
factor F
1
to help account for the natural tendency of gas forces
to exceed the values based directly on suction and discharge
pressures due to flow resistances and pulsations. Machines
with good pulsation control and low external flow resistance
may achieve F
1

as small as 1.1; for machines with low
compression ratio, high pulsations, or highly resistive flow
through piping and nozzles, F
1
can approach 1.5 or even
higher. A reasonable working value for F
1
is 1.15 to 1.2.
Preferably, the maximum rod force resulting from gas
pressures is based on knowledge of the continuous variation of
pressure in the cylinder (measured or predicted). In a repair
situation, measured cylinder pressure variation using a cylinder
analyzer provides the most accurate value of gas forces. Even
without cylinder pressure analysis, extreme operating values of
Fig. 3.4—Schematic of double-acting compressor cylinder
and piston.
351.3R-14 ACI COMMITTEE REPORT
suction and discharge pressure for each stage should be
recorded before the repair and used in the Eq. (3-22).
On new compressors, the engineer should ask the machine
manufacturer to provide values for maximum compressive
and tensile gas loads on each cylinder rod and, if these are
based on suction and discharge pressures, to recommend a
value of F
1
.
Gas forces act on the crankshaft with an equal and opposite
reaction on the cylinder. Thus, crankshaft and cylinder
forces globally balance each other. Between the crankshaft
and the cylinder, however, the compressor frame stretches or

contracts in tension or compression under the action of the
gas forces. The forces due to frame deflections are trans-
mitted to the foundation through connections with the
compressor frame. When acting without slippage, the frame
and foundation become an integral structure and together
stretch or contract under the gas loads.
The magnitude of gas force transferred into the foundation
depends on the relative flexibility of the compressor frame.
A very stiff frame transmits only a small fraction of the gas
force while a very flexible frame transmits most or all of the
force. Similar comments apply to the transfer of individual
cylinder inertia forces.
Based on limited comparisons using finite element analysis
(Smalley 1988), the following guideline is suggested for gas
and inertia force loads transmitted to the foundation by a
typical compressor
F
block
= F
rod
/F
red
(3-25)
(F
bolt
)
CHG
= [(F
rod
)/(N

bolt
)
CHG
]/F
red
(3-26)
(F
bolt
)
frame
= [(F
unbalance
/(N
bolt
)]/F
red
(3-27)
where
F
block
= the force acting outward on the block from
which concrete stresses should be calcu-
lated, lbf (N);
(F
bolt
)
CHG
= the force to be restrained by friction at the
cross head guide tie-down bolts, lbf (N);
(F

bolt
)
frame
= the force to be restrained by friction at the
frame tie-down bolts, lbf (N);
F
red
= a force reduction factor with suggested
value of 2, to account for the fraction of
individual cylinder load carried by the
compressor frame (“frame rigidity factor”);
(N
bolt
)
CHG
= the number of bolts holding down one
crosshead guide;
(N
bolt
)
frame
= the number of bolts holding down the
frame, per cylinder;
F
rod
= force acting on piston rod, from Eq. (3-22),
lbf (N); and
F
unbalance
= the maximum value from Eq. (3-18) applied

using parameters for a horizontal
compressor cylinder, lbf (N).
The factor F
red
is used to simplify a complex problem, thus
avoiding the application of unrealistically high loads on the
anchor bolts and the foundation block. The mechanics
involved in transmitting loads are complex and cannot easily
be reduced to a simple relationship between a few parameters
beyond the given load equations. A detailed finite-element
analysis of metal compressor frame, chock mounts, concrete
block, and grout will account for the relative flexibility of the
frame and its foundation in determining individual anchor
bolt loads and implicitly provide a value for F
red
. If the
frame is very stiff relative to the foundation, the value for
F
red
will be higher, implying more of the transmitted loads
are carried by the frame and less by the anchor bolts and
foundation block. Based on experience, a value of 2 for this
factor is conservatively low; however, higher values have
been seen with frames designed to be especially stiff.
Simplifying this approach, one report (Smalley and
Harrell 1997) suggests using a finite element analysis to
calculate forces transmitted to the anchor bolts. If a finite
element analysis is not possible, the engineer should get
from the machine manufacturer or calculate the maximum
horizontal gas force and maximum horizontal inertia force

for any throw or cylinder. The mounts, anchor bolts, and
blocks are then designed for
F
THROW
= (greater of F
GMAX
or F
IMAX
)/2 (3-28)
where
F
GMAX
= maximum horizontal gas force on a throw or
cylinder, lbf (N);
F
IMAX
= maximum horizontal inertia force on a throw or
cylinder, lbf (N); and
F
THROW
= horizontal force to be resisted by each throw’s
anchor bolts, lbf (N).
3.2.3.3 Reciprocating inertia loads for multicylinder
machines—As a practical matter, most reciprocating
machines have more than one cylinder, and manufacturers
arrange the machine components in a manner that minimizes
the net unbalanced forces. For example, rotating parts like
the crankshaft can be balanced by adding or removing
correcting weights. Translating parts like pistons and those
that exhibit both rotation and translation, like connecting

rods, can be arranged in such a way as to minimize the unbal-
anced forces and moments generated. Seldom, if ever, is it
possible to perfectly balance reciprocating machines.
The forces generated by reciprocating mechanisms are
functions of the mass, stroke, piston arrangement,
connecting rod size, crank throw orientation (phase angle),
and the mass and arrangement of counterweights on the
crankshaft. For this reason, calculating the reciprocating
forces for multicylinder machines can be quite complex and
are therefore normally provided by the machine manufac-
turer. If the machine is an integral engine compressor, it can
include, in one frame, cylinders oriented horizontally, verti-
cally, or in between, all with reciprocating inertias.
Some machine manufacturers place displacement transducers
and accelerometers on strategic points on the machinery. They
can then measure displacements and accelerations at those
points for several operational frequencies to determine the
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-15
magnitude of the unbalanced forces and couples for multi-
cylinder machines.
3.2.3.4 Estimating reciprocating inertia forces from
multicylinder machines—In cases where the manufacturer’s
data are unavailable or components are being replaced, the
engineer should use hand calculations to estimate the recip-
rocating forces from a multicylinder machine. One such
procedure for a machine having n number of cylinders is
discussed by Mandke and Troxler (1992). Section 3.7
summarizes this method.
3.2.4 Impulsive machine loads—The impulsive load
generated by a forging hammer is caused by the impact of the

hammer ram onto the hammer anvil. This impact process
transfers the kinetic energy of the ram into kinetic energy of
the entire hammer assembly. The post-impact velocity of the
hammer is represented by
(3-29)
where
v
h
= post-impact hammer velocity, ft/s (m/s);
M
r
= ram mass including dies and ancillary parts, lbm (kg);
M
h
= hammer mass including any auxiliary foundation, lbm
(kg);
α
h
= ram rebound velocity relative to impact velocity; and
v
r
= ram impact velocity, ft/s (m/s).
General experience indicates that α
h
is approximately
60% for many forging hammer installations. From that point,
the hammer foundation performance can be assessed as a
rigid body oscillating as a single degree-of-freedom system
with an initial velocity of v
h

.
For metal-forming presses, the dynamic forces develop
from two sources: the mechanical movement of the press
components and material-forming process. Each of these
forces is unique to the press design and application and needs
to be evaluated with proper information from the press
manufacturer and the owner.
The press mechanics often include rotating and recipro-
cating components. The dynamic forces from these indi-
vidual pieces follow the rules established in earlier sections
of this document for rotating and reciprocating components.
Only the press manufacturer familiar with all the internal
components can knowledgeably calculate the specific
forces. Figure 2.3 presents a horizontal force time-history for
a forging press. Similar presses can be expected to have
similar characteristics; however, the particular values and
timing data differ.
The press drive mechanisms include geared and direct-
drive systems. Depending on the design, these drives may or
may not be balanced. The press slide travels vertically
through a set stroke of 1/2 in. (12 mm) to several inches at a
given speed. Some small presses may have inclinable beds
so that the slide is not moving vertically. It is often adequate
to assume that the slide moves in a vertical path defined by a
circularly rotating crankshaft, that is
(3-30)
where
d
s
= displacement of the slide, in. (mm);

S = press stroke, in. (mm); and
ω
o
= circular operating frequency of the machine (rad/s).
This leads to a dynamic inertia force from the slide of
lbf (3-31)
N
where
F
s
= dynamic inertia force of slide, lbf (N); and
m
d
= slide mass including the effects of any balance mech-
anism, lbm (kg).
This assumption is based on simple circular motions and
simple linkages. Other systems may be in-place to increase
the press force and improve the timing. These other systems
may increase the acceleration of the unbalanced weights and
thus alter the magnitude and frequency components of the
dynamic force transmitted to the foundation.
3.2.5 Loading conditions—During their lives, machinery
equipment support structures and foundations undergo
different loading conditions including erection, testing, shut-
down, maintenance, and normal and abnormal operation. For
each loading condition, there can be one or more combinations
of loads that apply to the structure or foundation. The following
loading conditions are generally considered in design:
• Erection condition represents the design loads that act
on the structure/foundation during its construction;

• Testing condition represents the design loads that act on
the structure/foundation while the equipment being
supported is undergoing testing, such as hydrotest;
• Empty (shutdown) represents the design loads that act
on the structure when the supported equipment is at its
least weight due to removal of process fluids, applicable
internals, or both as a result of maintenance or other
out-of-service disruption;
• Normal operating condition represents the design loading
during periods of normal equipment operation; and
• Abnormal operating condition represents the design
loading during periods when unusual or extreme operating
loads act on the structure/foundation.
3.2.6 Load combinations—Table 3.2 shows the general
classification of loads for use in determining the applicable
load factors in strength design (ACI 318). In considering soil
stresses, the normal approach is working stress design
without load factors and with overall factors of safety iden-
tified as appropriate by geotechnical engineers. The load
combinations frequently used for the various load conditions
are as follows:
1. Erection
a) Dead load + erection forces
v
h
M
r
M
h
1 α

h
+()v
r
=
d
s
t()
S
2
ω
o
t()sin=
F
s
t() m
d
ω
o
2
S
2
ω
o
t()12⁄sin=
F
s
t() m
d
ω
o

2
S
2
ω
o
t()1000⁄sin=
351.3R-16 ACI COMMITTEE REPORT
b) Dead load + erection forces + reduced wind + snow,
ice, or rain
c) Dead load + erection forces + seismic + snow, ice, or rain
2. Testing
a) Dead load + test loads
b) Dead load + test loads + live + snow, ice, or rain
c) Dead load + test loads + reduced wind + snow, ice,
or rain
3. Empty (shutdown)
a) Dead load + maintenance forces + live load + snow,
ice, or rain
4. Normal operation
a) Dead load
b) Dead load + thermal load + machine forces + live
loads + wind + snow, ice, or rain
c) Dead load + thermal load + machine forces + seismic
+ snow, ice, or rain
5) Abnormal operation
a) Dead load + upset (abnormal) machine loads + live +
reduced wind
It is common to only use some fraction of full wind, such as
80% in combination with erection loads and 33% for test
loads, due to the short duration of these conditions (ASCE 7).

3.3—Dynamic soil properties
Soil dynamics deals with engineering properties and
behavior of soil under dynamic stress. For the dynamic analysis
of machine foundations, soil properties, such as Poisson’s
ratio, dynamic shear modulus, and damping of soil, are
generally required.
Though this work is typically completed by a geotechnical
engineer, this section provides a general overview of
methods used to determine the various soil properties. Many
references are available that provide a greater level of detail
on both theory and standard practice, including Das (1993),
Bowles (1996), Fang (1991), and Arya, O’Neill, and Pincus
(1979). Seed and Idriss (1970) provide greater detail on
items that influence different soil properties.
This section does not cover considerations that affect the
suitability of a given soil to support a dynamic machine
foundation. Problems could include excessive settlement
caused by dynamic or static loads, liquefaction, dimensional
stability of a cohesive soil, frost heave, or any other relevant
soils concern.
In general, problems involving the dynamic properties of
soils are divided into small and large strain amplitude
responses. For machine foundations, the amplitudes of
dynamic motion, and consequently the strains in the soil, are
usually low (strains less than10
–3
%). A foundation that is
subjected to an earthquake or blast loading is likely to
undergo large deformations and, therefore, induce large
strains in the soil. The information in this report is only

applicable for typical machine foundation strains. Refer to
Seed and Idriss (1970) for information on strain-related
effects on shear modulus and material damping.
The key soil properties, Poisson’s ratio and dynamic shear
modulus, may be significantly affected by water table vari-
ations. Prudence suggests that in determining these properties,
such variations be considered and assessed, usually in
conjunction with the geotechnical engineers. This approach
often results in expanding the range of properties to be
considered in the design phase.
3.3.1 Poisson’s ratio—Poisson’s ratio ν, which is the ratio
of the strain in the direction perpendicular to loading to the
strain in the direction of loading, is used to calculate both the
soil stiffness and damping. Poisson’s ratio can be computed
from the measured values of wave velocities traveling through
the soil. These computations, however, are difficult. The
stiffness and damping of a foundation system are generally
insensitive to variations of Poisson’s ratio common in soils.
Generally, Poisson’s ratio varies from 0.25 to 0.35 for
cohesionless soils and from 0.35 to 0.45 for cohesive soils. If
no specific values of Poisson’s ratio are available, then, for
design purposes, the engineer may take Poisson’s ratio as
0.33 for cohesionless soils and 0.40 for cohesive soils.
3.3.2 Dynamic shear modulus—Dynamic shear modulus
G is the most important soil parameter influencing the
dynamic behavior of the soil-foundation system. Together
with Poisson’s ratio, it is used to calculate soil impedance.
Refer to Section 4.2 for the discussion on soil impedance.
The dynamic shear modulus represents the slope of the
shear stress versus shear strain curve. Most soils do not

respond elastically to shear strains; they respond with a
combination of elastic and plastic strain. For that reason,
plotting shear stress versus shear strain results in a curve not
a straight line. The value of G varies based on the strain
considered. The lower the strain, the higher the dynamic
shear modulus.
Several methods are available for obtaining useful values
of dynamic shear modulus:
• Field measurements of stress wave velocities of in-place
soils;
• Laboratory tests on soil samples; and
• Correlation to other soil properties.
Due to variations inherent in the determination of dynamic
shear modulus values, it may be appropriate to complete more
than one foundation analysis. One analysis could be
completed with the minimum possible value, one could be
completed using the maximum possible value, and then addi-
tional analyses could be completed with intermediate values.
Table 3.2—Load classifications for ultimate
strength design
Design loads
Load
classification
Weight of structure, equipment, internals, insulation, and
platforms
Temporary loads and forces caused by erection
Fluid loads during testing and operation
Thermal loads
Anchor and guide loads
Dead

Platform and walkway loads
Materials to be temporarily stored during maintenance
Materials normally stored during operation such as tools
and maintenance equipment
Vibrating equipment forces
Impact loads for hoist and equipment handling utilities
Live
Earthquake loads
Transportation loads
Snow, ice, or rain loads
Wind loads
Environmental
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-17
3.3.2.1 Field determination—Field measurements are
the most common method for determining the dynamic shear
modulus of a given soil. These methods involve measuring
the soil characteristics, in-place, as close as possible to the
actual foundation location(s).
Because field determinations are an indirect determination
of shear modulus, the specific property measured is the shear
wave velocity. There are three different types of stress waves
that can be transmitted through soil or any other elastic body.
• Compression (primary P) waves;
• Shear (secondary S) waves; and
• Rayleigh (surface) waves.
Compression waves are transmitted through soil by a
volume change associated with compressive and tensile
stresses. Compression waves are the fastest of the three
stress waves.
Shear waves are transmitted through soil by distortion

associated with shear stresses in the soil and are slower than
compression waves. No volume change occurs in the soil.
Rayleigh waves occur at the free surface of an elastic body;
typically, this is the ground surface. Rayleigh waves have
components that are both perpendicular to the free surface
and parallel to the free surface and are slightly slower than
shear waves.
Several methods are available for measuring wave veloci-
ties of the in-place soil:
• The cross-hole method;
• The down-hole method;
• The up-hole method; and
• Seismic reflection (or refraction).
In the cross-hole method, two vertical boreholes are
drilled. A signal generator is placed in one hole and a sensor
is placed in the other hole. An impulse signal is generated in
one hole, and then the time the shear wave takes to travel
from the signal generator to the sensor is measured. The
travel time divided by the distance yields the shear wave
velocity. The cross-hole method can be used to determine G
at different depths (Fig. 3.5).
In the down-hole method, only one vertical borehole is
drilled. A signal generator is placed at the ground surface
some distance away from the borehole, and a sensor is placed
in the bottom of the borehole. An impulse signal is generated,
and then the time the shear wave takes to travel from the
signal generator to the sensor is measured. The travel time
divided by the distance yields the shear wave velocity. This
method can be run several different times, with the signal
generator located at different distances from the borehole

each time. This permits the measuring of soil properties at
several different locations, which can then be averaged to
determine an average shear wave velocity (Fig. 3.6).
The up-hole method is similar to the down-hole method.
The difference is that the signal generator is placed in the
borehole and the sensor is placed at the ground surface.
Dynamic shear modulus and measured-in-field shear
wave velocity are related as follows
G = ρ(V
s
)
2
(3-32)
where
G = dynamic shear modulus of the soil, lbf/ft
2
(Pa);
V
s
= shear wave velocity of the soil, ft/s (m/s); and
ρ = soil mass density, lbm/ft
3
(kg/m
3
).
An alternative field method is to use reflection or refraction
of elastic stress waves. These methods are based on the
principle that when elastic waves hit a boundary between
dissimilar layers, the wave is reflected or refracted. This
method should only be used at locations where the soils are

deposited in discrete horizontal, or nearly horizontal layers,
or at locations where soil exists over top of bedrock. This
method consists of generating a stress wave at one location
at the ground surface and measuring the time it takes for the
stress wave to reach a second location at the ground surface.
The wave travels from the ground surface to the interface
between differing soils layers, travels along the interface,
then back to the ground surface. The time the wave takes to
travel from the signal generator to the sensor is a function of
the soils properties and the depth of the soil interface. One
advantage of this method is that no boreholes are required.
Also, this method yields an estimated depth to differing soil
layers. One disadvantage is that this method cannot be used
when the groundwater table is near the ground surface.
3.3.2.2 Laboratory determination—Laboratory tests are
considered less accurate than field measurements due to the
Fig. 3.5—Schematic of cross-hole technique.
Fig. 3.6—Equipment and instrumentation for down-hole
survey.
351.3R-18 ACI COMMITTEE REPORT
possibility of sample disturbance. Sometimes laboratory
tests are used to validate field measurements when a high
level of scrutiny is required, for instance, when soil properties
are required for a nuclear energy facility.
The most common laboratory test is the Resonant-Column
method, where a cylindrical sample of soil is placed in a
device capable of generating forced vibrations. The soil
sample is exited at different frequencies until the resonant
frequency is determined. The dynamic soil modulus can be
calculated based on the frequency, the length of the soil

sample, the end conditions of the soil sample, and the density
of the soil sample. ASTM D 4015 defines the Resonant-
Column method.
3.3.2.3 Correlation to other soil properties—Correlation
is another method for determining dynamic soils properties.
The engineer should be careful when using any correlation
method because these are generally the least-accurate
methods. The most appropriate time to consider using these
methods is for preliminary design or for small noncritical
applications with small dynamic loads. Correlation to other
soil properties should be considered as providing a range of
possible values, not providing a single exact value.
Hardin and Richart (1963) determined that soil void ratio
e
v
and the probable confining pressure σ
o
had the most
impact on the dynamic shear modulus. Hardin and Black
(1968) developed the following relationships:
For round-grained sands with e < 0.8, dynamic shear
modulus can be estimated from
lbf/ft
2
(3-33)
Pa
For angular-grained materials with e > 0.6 and normally
consolidated clays with low surface activity, dynamic shear
modulus can be estimated from
lbf/ft

2
(3-34)
Pa
G
31,530 2.17 e
v
–()
2
σ
o
1 e
v
+
=
G
218,200 2.17 e
v
–()
2
σ
o
1 e
v
+
=
G
14,760 2.97 e
v
–()
2

σ
o
1 e
v
+
=
G
102,140 2.97 e
v
–()
2
σ
o
1 e
v
+
=
In the previous equations,
e
v
= void ratio; and
σ
o
= probable confining pressure, lbf/ft
2
(Pa).
In general, relative density in sand is proportional to the
void ratio. Seed and Idriss (1970) provide guidance for
correlating the dynamic shear modulus to relative density in
sand, along with the confining pressure

lb/ft
2
(3-35)
Pa
where K
2
= a parameter that depends on void ratio and strain
amplitude. Table 3.3 provides values of K
2
with respect to
relative density.
3.3.3 Damping of soil—Damping is a phenomenon of
energy dissipation that opposes free vibrations of a system.
Like the restoring forces, the damping forces oppose the
motion, but the energy dissipated through damping cannot
be recovered. A characteristic feature of damping forces is
that they lag the displacement and are out of phase with the
motion. Damping of soil includes two effects—geometric
and material damping.
Geometric, or radiation, damping reflects energy dissipation
through propagation of elastic waves away from the immediate
vicinity of a foundation and inelastic deformation of soil. It
results from the practical infinity of the soil medium, and it
is close to viscous in character. Refer to Chapter 4 for
methods of computing geometric damping.
Material, or hysteretic, damping reflects energy dissipation
within the soil itself due to the imperfect elasticity of real
materials, which exhibit a hysteric loop effect under cyclic
loading (Fig. 3.7). The amount of dissipated energy is given
by the area of the hysteretic loop. The hysteretic loop implies

a phase shift between the stress and strain because there is a
stress at zero strain and vice versa, as can be seen from Fig. 3.7.
The amount of dissipated energy depends on strain
(displacement) but is essentially independent of frequency,
as shown on Fig. 3.8.
The magnitude of material damping can be established
experimentally using the hysteretic loop and the relation
(3-36)
where
β
m
= material damping ratio;
∆W = area enclosed by the hysteretic loop; and
W = strain energy.
Instead of an experimental determination, many practitioners
use a material damping ratio of 0.05, or 5%. The material
damping ratio is fairly constant for small strains but increases
with strain due to the nonlinear behavior of soils.
The term material or hysteretic damping implies
frequency independent damping. Experiments indicate that
frequency independent hysteretic damping is much more
G 1000K
2
σ
o
=
G 6920K
2
σ
o

=
β
m
1
4π

∆W
W
=
Table 3.3—Values of K
2
versus relative density
(Seed and Idriss 1970)
Relative density, %
K
2
90 70
75 61
60 52
45 43
40 40
30 34
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-19
typical of soils than viscous damping because the area of the
hysteretic loop does not grow in proportion to the frequency.
3.4—Vibration performance criteria
The main purposes of the foundation system with respect
to dynamic loads include limiting vibrations, internal loads,
and stresses within the equipment. The foundation system
also limits vibrations in the areas around the equipment

where other vibration-sensitive equipment may be installed,
personnel may have to work on a regular basis, or damage to
the surrounding structures may occur. These performance
criteria are usually established based on vibration amplitudes
at key points on or around the equipment and foundation
system. These amplitudes may be based on displacement,
velocity, or acceleration units. Displacement limitations are
commonly based on peak-to-peak amplitudes measured in
mils (0.001 in.) or microns (10
–6
m). Velocity limitations are
typically based on either peak velocities or root-mean-square
(rms) velocities in units of inch per second or millimeter per
second. Displacement criteria are almost always frequency
dependent with greater motions tolerated at slower speeds.
Velocity criteria may depend on frequency but are often
independent. Acceleration criteria may be constant with
frequency or dependent.
Some types of equipment operate at a constant speed while
other types operate across a range of speeds. The foundation
engineer should consider the effect of these speed variations
during the foundation design.
3.4.1 Machine limits—The vibration limits applicable to
the machine are normally set by the equipment manufacturer
or are specified by the equipment operator or owner. The
limits are usually predicated on either limiting damage to the
equipment or ensuring proper performance of the equipment.
Limits specified by operators of the machinery and design
engineers are usually based on such factors as experience or
the installation of additional vibration monitoring equipment.

For rotating equipment (fans, pumps, and turbines), the
normal criterion limits vibration displacements or velocities at
the bearings of the rotating shaft. Excessive vibrations of the
bearings increase maintenance requirements and lead to
premature failure of the bearings. Often, rotating equipment
has vibration switches to stop the equipment if vibrations
become excessive.
Reciprocating equipment (diesel generators, compressors,
and similar machinery) tends to be more dynamically rugged
than rotating equipment. At the same time, it often generates
greater dynamic forces. While the limits may be higher,
motions are measured at bearing locations. In addition, opera-
tors of reciprocating compressors often monitor vibrations of
the compressor base relative to the foundation (sometimes
called “frame movement”) as a measure of the foundation and
machine-mounting condition and integrity.
Impulsive machines (presses, forging hammers) tend not to
have specific vibration limitations as controllable by the foun-
dation design. With these machines, it is important to recognize
the difference between the inertial forces and equipment
dynamics as contrasted with the foundation system dynamics.
The forces with the equipment can generate significant
accelerations and stresses that are unrelated to the stiffness,
mass, or other design aspect of the foundation system. Thus,
monitoring accelerations in particular on an equipment frame
may not be indicative of foundation suitability or adequacy.
Researchers have presented various studies and papers
addressing the issues of machinery vibration limits. This
variety is reflected in the standards of engineering companies,
plant owners, and industry standards. When the equipment

manufacturer does not establish limits, recommendation from
ISO 10816-1, Blake (1964), and Baxter and Bernhard (1967)
are often followed. Most of these studies relate directly to
rotating equipment. In many cases they are also applicable to
reciprocating equipment. Rarely do these studies apply to
impulsive equipment.
ISO publishes ISO 10816 in a series of six parts to address
evaluation of machinery vibration by measurements on the
nonrotating parts. Part 1 provides general guidelines and sets
the overall rules with the subsequent parts providing specific
values for specific machinery types. These standards are
primarily directed toward in-place measurements for the
assessment of machinery operation. They are not intended to
identify design standards. Design engineers, however, have
used predecessor documents to ISO 10816 as a baseline for
design calculations and can be expected to do similarly with
these more recent standards.
The document presents vibration criteria in terms of rms
velocity. Where there is complexity in the vibration signal
(beyond simple rotor unbalance), the rms velocity basis
provides the broad measure of vibration severity and can be
correlated to likely machine damage. For situations where
the pattern of motion is fairly characterized by one simple
harmonic, such as simple rotor unbalance, the rms velocities
can be multiplied by √2 to determine corresponding peak
velocity criteria. For these same cases, displacements can be
calculated as
Fig. 3.7—Hysteretic loop.
Fig. 3.8—Comparison of viscous and hysteretic damping.
351.3R-20 ACI COMMITTEE REPORT

(3-37)
where
v = displacement amplitude, in. (cm);
v′ = velocity, in./s (cm/s);
ω
m
= circular frequency of motion, rad/s; and
f
m
= frequency of motion, Hz.
The rms velocity yields an rms displacement, and a peak
velocity results in a zero-to-peak displacement value, which
can be doubled to determine a peak-to-peak displacement
v
v′
ω
m

v′
2πf
m
==
value. If the motion is not a simple pure harmonic motion, a
simple relationship among the rms displacement, rms velocity,
peak velocity, zero-to-peak displacement, and peak-to-peak
displacement does not exist.
ISO 10816-1 identifies four areas of interest with respect
to the magnitude of vibration measured:
• Zone A: vibration typical of new equipment;
• Zone B: vibration normally considered acceptable for

long-term operation;
• Zone C: vibration normally considered unsatisfactory
for long-term operation; and
• Zone D: vibration normally considered severe enough
to damage the machine.
The subsequent parts of ISO 10816 establish the boundaries
between these zones as applicable to specific equipment.
Part 2, ISO 10816-2, establishes criteria for large, land-
based, steam-turbine generator sets rated over 67,000 horse-
power (50 MW). The most general of the standards is Part 3,
ISO 10816-3, which addresses in-place evaluation of general
industrial machinery nominally more than 15 kW and operating
between 120 and 15,000 rpm. Within ISO 10816-3, criteria are
established for four different groups of machinery, and
provisions include either flexible or rigid support conditions.
Criteria are also established based on both rms velocity and rms
displacement. Part 4, ISO 10816-4, identifies evaluation criteria
for gas-turbine-driven power generation units (excluding
aircraft derivatives) operating between 3000 and 20,000 rpm.
Part 5 (ISO 10816-5) applies to machine sets in hydro-power
facilities and pumping plants. Part 6, ISO 10816-6, provides
evaluation criteria for reciprocating machines with power
ratings over 134 horsepower (100 kW). The scope of Part 5 is
not applicable to general equipment foundations and the
criteria of Part 6 are not sufficiently substantiated and defined
to be currently useful.
Another document available for establishing vibration
limitation is from Lifshits (Lifshits, Simmons, and Smalley
1986). This document follows Blake’s approach of identi-
fying five different categories from No Faults to Danger of

Immediate Failure. In addition, a series of correction factors
are established to broaden the applicability to a wider range
of equipment and measurement data.
Blake’s paper (Blake 1964) has become a common basis
for some industries and firms. His work presented a standard
vibration chart for process equipment with performance
rated from “No Faults (typical of new equipment)” to
“Dangerous (shut it down now to avoid danger).” The chart
was primarily intended to aid plant personnel in assessing field
installations and determining maintenance plans. Service
factors for different types of equipment are used to allow
widespread use of the basic chart. This tool uses vibration
displacement (in. or mm) rather than velocity and covers speed
ranges from 100 to 10,000 rpm. Figure 3.9 and Table 3.4
present the basic chart and service factors established by Blake.
Baxter and Bernhard (1967) offered more general vibration
tolerances in a paper that has also become widely referenced.
Again with primary interest to the plant maintenance operations,
they established the General Machinery Vibration Severity
Chart, shown in Fig. 3.10, with severity ranging from extremely
Table 3.4—Service factors from Blake (Richart,
Hall, and Woods 1970)
Item
Bolted
down
Not
bolted
down
Single-stage centrifugal pump, electric motor, fan 1.0 0.4
Typical chemical processing equipment, noncritical 1.0 0.4

Turbine, turbo-generator, centrifugal compressor 1.6 0.6
Centrifugal, stiff-shaft (at basket housing), multi-stage
centrifugal pump
2.0 0.8
Miscellaneous equipment, characteristics unknown 2.0 0.8
Centrifuge, shaft-suspended, on shaft near basket 0.5 0.2
Centrifuge, link-suspended, slung 0.3 0.1
Notes: 1. Vibration is measured at the bearing housing except as noted; 2. Machine
tools are excluded; and 3) Compared or measured displacements are multiplied by the
appropriate service factor before comparing with Fig. 3.9.
Fig. 3.9—Vibration criteria for rotating machinery (Blake
1964, as modified by Arya, O’Neill, and Pincus 1979).
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-21
smooth to very rough. These are plotted as displacement versus
vibration frequency so that the various categories are differenti-
ated along lines of constant peak velocity.
The American Petroleum Institute (API) also has a series
of standards for equipment common in the petrochemical
industry (541, 610, 612, 613, 617, 618, and 619). ISO 10816-3
can be applied for some large electrical motors; however,
most design offices do not generally perform rigorous analyses
for these items.
Figure 3.11 provides a comparison of five generic standards
against four corporate standards. To the extent possible, the
comparisons are presented on a common basis. In particular,
the comparison is based on equipment that is in service,
perhaps with minor faults, but which could continue in
service indefinitely. The Blake line is at the upper limit of the
zone identifying operation with minor faults with a service
factor of one applicable for fans, some pumps, and similar

equipment. The Lifshits line separates the acceptable and
marginal zones and includes a K of 0.7, reflecting equipment
with rigid rotors. The ISO lines are drawn at the upper level
of Zone B, normally considered acceptable for long-term
operation. The ISO 10816-3 line is for large machines
between 400 and 67,000 horsepower (300 kW and 50 MW)
on rigid support systems. The ISO 10816-2 is for large
turbines over 67,000 horsepower (50 MW).
The company standards are used for comparison to calculate
motions at the design stage. For these calculations, the compa-
nies prescribe rotor unbalance conditions worse than those
expected during delivery and installation. These load defini-
tions are consistent with those presented in Section 3.2.2.1.
Thus, there is a level of commonality. Company G’s criteria
are for large turbine applications and, thus, most comparable
to the ISO 10816-2 criteria. The other company standards are
for general rotating equipment. Company F permits higher
motions for reciprocating equipment. In all cases the design
companies standards reflect that the manufacturer may
establish equipment-specific criteria that could be more
limiting than their internal criteria.
Figure 3.11 shows that the corporate standards are generally
below the generic standards because the generic standards are
intended for in-place service checks and maintenance decisions
rather than offering initial design criteria. One company is
clearly more lenient for very low-speed equipment, but the
corporate standards tend to be similar.
The Shock and Vibration Handbook (Harris 1996)
contains further general information on such standards.
3.4.2 Physiological limits—Human perception and sensi-

tivity to vibration is ambiguous and subjective. Researchers
have studied and investigated this topic, but there are no clear
uniform U.S. standards. In Germany, VDI 2057 provides
guidance for the engineer. Important issues are the personnel
expectations and needs and the surrounding environment.
ISO 2631 provides guidance for human exposure to
whole-body vibration and considers different comfort levels
and duration of exposure. This document does not address
the extensive complexities identified in ISO 2631. Figure 3.12
presents the basic suggested acceleration limits from ISO
2631 applicable to longitudinal vibrations (vertical for a
standing person). This figure reflects the time of exposure and
frequency consideration for fatigue-decreased proficiency.
The figure shows that people exhibit fatigue and reduced profi-
ciency when subjected to small accelerations for long periods or
greater accelerations for shorter periods. The frequency of the
accelerations also impact fatigue and proficiency.
The modified Reiher-Meister figure (barely perceptible,
noticeable, and troublesome) is also used to establish limits
with respect to personnel sensitivity, shown in Fig. 3.13.
Fig. 3.10—General Machinery Vibration Severity Chart
(Baxter and Bernhard 1967).
Fig. 3.11—Comparison of permissible displacements.
351.3R-22 ACI COMMITTEE REPORT
DIN 4150 is another standard used internationally. Part 3
defines permissible velocities suitable for assessment of short-
term vibrations on structures, which are given in Table 3.5.
Furthermore, Part 2 of this German standard defines limitations
for allowable vibrations based on perception as a function of
location (residential, light industrial) and either daytime or

nighttime. Most engineering offices do not consider human
perception to vibrations, unless there are extenuating circum-
stances (proximity to office or residential areas).
There are no conclusive limitations on the effects of vibration
of surrounding buildings. The Reiher-Meister figure identifies
levels of vibration from mining operations that have
damaged structures.
3.4.3 Frequency ratios—The frequency ratio is a term that
relates the operating speed of the equipment to the natural
frequencies of the foundation. Engineers or manufacturers
require that the frequency of the foundation differ from the
operating speed of the equipment by certain margins. This
limitation is applied to prevent resonance conditions from
developing within the dynamic soil-foundation-equipment.
The formulation or presentation of frequency ratios may be
based around either f
o
/f
n
or f
n
/f
o
(operating frequency to
natural frequency or its inverse), and engineers or manufac-
turers should exercise caution to prevent misunderstandings.
A common practice among engineering firms is to
compute the natural frequencies of the basic equipment-
foundation and compare the values with the dynamic excitation
frequency. Many companies require that the natural

frequency be 20 to 33% removed from the operating speed.
Some firms have used factors as low as 10% or as high as
50%. If the frequencies are well separated, no further evaluation
is needed. If there is a potential for resonance, the engineer
should either adjust to the foundation size or perform more
refined calculations. Refined calculations may include an
analysis with a deliberately reduced level of damping. The
size and type of equipment play an important role in this
decision process.
Frequency ratio is a reasonable design criterion, but one
single limiting value does not fit all situations. Where there
is greater uncertainty in other design parameters (soil stiffness,
for example), more conservatism in the frequency ratio may
be appropriate. Similarly, vibration problems can exist even
though resonance is not a problem.
3.4.4 Transmissibility—A common tool for the assessment
of vibrations at the design stage is a transmissibility ratio, as
shown in Fig. 3.14, which is based on a single degree-of-
freedom (SDOF) system with a constant speed f
o
excitation
force. This ratio identifies the force transmitted through the
spring-damper system with the supporting system as
compared with the dynamic force generated by the equip-
ment. This ratio should be as low as possible, that is, only
transmit 20% of the equipment dynamic force into the
supporting system. Low transmissibility implies low vibra-
tions in the surroundings, but this is not an absolute truth.
This transmissibility figure assumes that the damping force is
directly and linearly proportional to the velocity of the SDOF.

Where the system characteristics are such that the damping
force is frequency dependent, the aforementioned represen-
Fig. 3.12—Longitudinal acceleration limits (adapted from
ISO 2631-1).
Fig. 3.13—Reiher-Meister Chart (Richart, Hall, and
Woods 1970).
Table 3.5—Short-term permissible values
(DIN 4150-3)
Type
of building
Foundation
(1 to 10 Hz)
Foundation
(10 to 50 Hz)
Foundation
(50 to 100 Hz)
Top complete
floor (all
frequencies)
Industrial and
commercial
0.8 in./s
(20 mm/s)
0.8 to 1.6 in./s
(20 to 40 mm/s)
1.6 to 2.0 in.s
(40 to 50 mm/s)
1.6 in./s
(40 mm/s)
Residential

0.2 in./s
(5 mm/s)
0.2 to 0.6 in./s
(5 to 15 mm/s)
0.6 to 0.8 in./s
(15 to 20 mm/s)
0.6 in./s
(15 mm/s)
Special or
sensitive
0.1 in./s
(3 mm/s)
0.1 to 0.3 in./s
(3 to mm/s)
0.3 to 0.4 in./s
(8 to 10 mm/s)
0.3 in./s
(8 mm/s)
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-23
tation is not accurate. When the damping resistance
decreases at higher frequencies, the deleterious effect of
damping on force transmissibility can be mitigated.
For soil or pile-supported systems, the transmissibility
ratio may not be meaningful. In SDOF models of these
systems, the spring and damper are provided by the soil and,
while the transmissibility of the design may be low, the
energy worked through these system components is motion
in the surroundings that may not be acceptable.
3.5—Concrete performance criteria
The design of the foundation should withstand all applied

loads, both static and dynamic. The foundation should act in
unison with the equipment and supporting soil or structure to
meet the deflection limits specified by the machinery manu-
facturer or equipment owner.
The service life of a concrete foundation should meet or
exceed the anticipated service life of the equipment installed
and resist the cyclic stresses from dynamic loads. Cracking
should be minimized to ensure protection of reinforcing steel.
The structural design of all reinforced concrete foundations
should be in accordance with ACI 318. The engineer may use
allowable stress methods for nonprestressed reinforced concrete.
In foundations thicker than 4 ft (1.2 m), the engineer may
use the minimum reinforcing steel suggested in ACI 207.2R.
API and the Construction Industry Institute published API
Recommended Practice 686/PIP REIE 686, “Recommended
Practice for Machinery Installation and Installation Design.”
Chapter 4 of 686/PIP REIE 686 includes design criteria for
soil-supported reinforced concrete foundations that supports
general and special purpose machinery. The concrete used in
the foundation should tolerate its environment during place-
ment, curing, and service. The engineer should consider
various exposures such as freezing and thawing, salts of
chlorides and sulfates, sulfate soils, acids, carbonation,
repeated wetting and drying, oils, and high temperatures.
In addition to conventional concrete, there are many tech-
nologies available—such as admixtures, additives, specialty
cements, and preblended products—to help improve place-
ment, durability, and performance properties. These addi-
tives include water reducers, set-controlling mixtures,
shrinkage-compensating admixtures, polymers, silica fumes,

fly ash, and fibers.
Many foundations, whether new or repaired, require a fast
turnaround to increase production by reducing downtime
without compromising durability and required strength.
These systems may use a combination of preblended or
field-mixed concrete and polymer concrete or grout to
reduce downtime to 12 to 72 h, depending on foundation
volume and start-up strength requirements.
3.6—Performance criteria for machine-
mounting systems
The machine-mounting system (broadly categorized as
either an anchorage-type or an isolator-type) attaches the
dynamic machine to its foundation. It represents a vital
interface between the machine and the foundation; however,
it can suffer from insufficient attention to critical detail by
the foundation engineer and machinery engineer because it
falls between their areas of responsibility. Anchorage-type
machine-mounting systems integrate the foundation and the
machine into a single structure. Isolator-type machine-
mounting systems separate the machine and the foundation
into two separate systems that may still dynamically interact
with each other. In the processes of design, installation, and
operation, the critical role of both types needs advocacy and
the assurance that interface issues receive attention. The
research and development of information on machine-
mounting system technology by the Gas Machinery
Research Council (GMRC) during the 1990s reflects the
importance that this group attaches to the anchorage-type
machine-mounting system. This research produced a series
of reports on machine-mounting topics (Pantermuehl and

Smalley 1997a,b; Smalley and Pantermuehl 1997; Smalley
1997). These reports, readily retrievable from www.gmrc.org,
are essential for those responsible for dynamic machines and
their foundations.
Most large machines, in spite of careful design for integrity
and function by their manufacturers, can internally absorb no
more than a fraction of the forces or thermal growth inherent
in their function.
Those responsible for the machine-foundation interface
should provide an attachment that transmits the remaining
forces for dynamic integrity of the structure yet accommo-
dates anticipated differential thermal expansion between
machine and foundation. They should recognize the inherent
conflicts in these requirements, the physical processes that
can inhibit performance of these functions, and the lifetime
constraints (such as limited maintenance and contaminating
materials) from which any dynamic machine can suffer as it
contributes to profitable, productive plant operation.
A dynamic machine may tend to get hotter and grow more
than its foundation (in the horizontal plane). The growth can
reach several tenths of an inch (0.1 in. = 2.54 mm); combustion
turbine casings grow so much that they have to include deliber-
ately installed flexibility between hotter and cooler elements
of their own metallic structure. Most machines—such as
compressors, steam turbines, motors, and generators—do
not internally relieve their own thermal growth, so the
mounting system should allow for thermal growth. Thermal
growth can exert millions of pounds of force (1,000,000 lbf
= 4500 kN), a level that cannot be effectively restrained.
Fig. 3.14—Force transmissibility.

351.3R-24 ACI COMMITTEE REPORT
Heat is transferred between the machine and foundation
through convection, radiation, and conduction. While
convection and radiation dominate in the regions where an
air gap separates the machine base from the foundation, the
mounting system provides the primary path for conduction.
Ten critical performance criteria can be identified as
generally applicable to isolator and anchorage-type
mounting systems:
1. A machine-mounting system should tolerate expected
differential thermal growth across the interface. This can
occur by combining strength to resist expansion forces and
stresses, flexibility to accommodate the deflections, and
tolerance for relative sliding across the interface (as the
machine grows relative to the foundation);
2. A machine-mounting system should either absorb or
transmit, across the mounting interface, those internally
generated dynamic forces, resulting from the machine’s
operation not absorbed within the machine structure itself.
These forces include both vertical and horizontal compo-
nents. Flexible mounts that deflect rather than restrain the
forces become an option only in cases where the machine
and any rigidly attached structure have the structural rigidity
needed to avoid damaging internal stresses and deflections.
Large machinery may not meet this criterion;
3. A nominally rigid mount should transmit dynamic
forces with only microlevel elastic deformation and negligible
dynamic slippage across the interface. The dynamic forces
should include local forces, such as forces from each individual
cylinder, which large machines transmit to the foundation

because of their flexibility. For reciprocating compressors,
this criterion helps ensure that the foundation and machine
form an integrated structure;
4. A machine-mounting system should perform its function
for a long life—typically 25 years or more. Specifications
from the operator should include required life;
5. Any maintenance and inspection required to sustain
integrity of the machine-mounting system should have a
frequency acceptable to the operator of the machine, for
example, once per year. Engineers, installers, and operators of
the machine and its foundation should agree to this mainte-
nance requirement because the design integrity relies on the
execution of these maintenance functions with this frequency;
6. The bolts that tie the machine to the mounting system,
and which form an integral part of the mounting system,
should have sufficient stretch and create enough normal
force across all interfaces to meet the force transmission and
deflection performance stated above;
7. The anchor bolt material strength should tolerate the
resultant bolt tensile stresses. The mounts, soleplates, and
grout layers compressed by the anchor bolt should tolerate
the compressive stresses imposed on them;
8. Any polymeric material (grout or chocks) compressed
by the anchor bolts should exhibit a tolerably low amount of
creep to maintain bolt stretch over the time period between
maintenance actions performed to inspect and tighten anchor
bolts. Indeed, the machine mount should perform its func-
tion, accounting for expected creep, even if maintenance
occurs less frequently;
9. The mounting system should provide a stable platform

from which to align the machine. Any deflections of the
mounting system that occur should remain sufficiently
uniform at different points to preserve acceptable alignment
of the machine. The specifications and use of adjustable
chock mounts has become increasingly widespread to
compensate for loss of alignment resulting from creep and
other permanent deformations; and
10. The mounting system should impose tolerable loads,
stresses, and deformations on the foundation itself. Appro-
priate foundation design to make the loads, stresses, and
deformations tolerable remains an essential part of this
performance criterion. Some of the loads and stresses to
consider include:
• Tensile stresses in the concrete at the anchor bolt termi-
nation point, which may cause cracks;
• Shear stresses in concrete above anchor bolt termination
points, which, if high enough, might result in pullout;
• Interface shear stresses between a grout layer and the
concrete resulting from the typically higher expansion
of polymer grout than concrete (best accommodated
with expansion joints); and
• Hogging or sagging deformation of the concrete block
produced by heat conduction through the mounting
system. Air gaps and low conductivity epoxy chocks
help minimize such deformation.
Potential conflicts requiring attention and management in
these performance criteria include:
• Requirements to accommodate thermal expansion
while transmitting dynamic forces; and
• Requirements to provide a large anchor bolt clamping

force (so that slippage is controlled during transfer of
high lateral loads) while stresses and deflections in bolt,
foundation, chocks, and grout remain acceptably low.
Physical processes that can influence the ability of the
mount to meet its performance criteria include:
• Creep—Creep of all polymeric materials under compres-
sive load. (Creep means time-dependent deflection
under load. Deflection increases with time—sometimes
doubling or tripling the initial deflection);
• Differential thermal expansion—This can occur when
two adjacent components at similar temperatures have
different coefficients of thermal expansion, when two
adjacent components of similar coefficients have different
temperatures or a combination of both. Machine
mounts with epoxy materials can experience both types
of differential thermal expansion;
• Friction—Friction is limited by a friction coefficient.
Friction defines the maximum force parallel to an
interface that the interface can resist before sliding for a
given normal force between the two interfacing materials;
• Limits on friction—The presence of oil in the interface
(typically cutting the dry friction coefficient in half)
causes further limits on friction;
• Yield strength—Yield strength of anchor bolts limits the
tension available from an anchor bolt and encourages
the use of high-strength anchor bolts for all critical
applications;
FOUNDATIONS FOR DYNAMIC EQUIPMENT 351.3R-25
• Cracks—Concrete can crack under tensile loads, and
these cracks can grow with time; and

• Oil—Oil can pool around many machinery installations.
Oil aggravates cracks in concrete, particularly under
alternating stresses where it induces a hydraulic action. In
many cases, oil, its additives, or the ambient materials it
transports react with concrete to reduce its strength, par-
ticularly in cracks where stresses tend to be high.
Those responsible for machine mounts, as part of a founda-
tion, should consider the aforementioned performance criteria,
the conflicts that complicate the process of meeting those
criteria, and the physical processes that inhibit the ability of any
installation to meet the performance criteria. Other sections of
this document address the calculation of loads, stresses, deflec-
tions, and the specific limits of strength implicit in different
materials. The GMRC reports referred to address all these
issues as they pertain to reciprocating compressors.
3.7—Method for estimating inertia forces
from multicylinder machines
The local horizontal F
zi
and vertical F
yi
unbalanced forces
for the i-th cylinder located in the horizontal plane can be
written as
F
zi
= (m
rec,i
+ m
rot,i

)r
i
ω
2
o
cos(ω
o
t + α
i
) (3-38)
+ m
rot,i
cos2(ω
o
t + α
i
)
and
(3-39)
where
m
rec,i
= reciprocating mass for the i-th cylinder;
m
rot,i
= rotating mass of the i-th cylinder;
r
i
= radius of the crank mechanism of the i-th cylinder;
L

i
= length of the connecting rod of the crank mecha-
nism at the i-th cylinder;
ω
o
= circular operating frequency of the machine (rad/s);
t = time, s; and
α
i
= the phase angle for the crank radius of the i-th
cylinder, rad.
The primary and secondary force components are as
follows
(primary)
F
zi
′ = (m
rec,i
+ m
rot,i
)r
i
cos(ω
o
t + α
i
) (3-40)
F
yi
′ = m

rec,i
r
i
sin(ω
o
t + α
i
) (3-41)
(secondary)
F
zi
′′ = (m
rot,i
)r
i
cos2(ω
o
t + α
i
) (3-42)
If the i-th cylinder is oriented at angle θ
i
to a global horizontal
z-axis, then the primary and secondary force components,
with respect to the global axis, can be rewritten as follows
(primary)
(3-43)
(3-44)
(secondary)
(3-45)

(3-46)
The resultant forces due to n cylinders in global coordinates
can be calculated as follows
(3-47)
(3-48)
(3-49)
(3-50)
The resultant moments due to n cylinders in global coordi-
nates can be determined as follows
(moment about y [vertical] axis) (3-51)
(3-52)
(moment about z [horizontal] axis) (3-53)
r
i
2
ω
o
2
L
i

F
yi
m
rec i,
r
i
ω
o
2

ω
o
t α
i
+()sin=
ω
o
2
ω
o
2
r
i
2
ω
o
2
L
i

F
zi
GP
F′
zi
θ
i
cos F′
yi
θ

i
sin==
F
yi
GP
F′
zi
θ
i
sin F′
yi
θ
i
cos==
F
zi
GS
F′′
zi
θ
i
cos=
F
yi
GS
F′′
zi
θ
i
sin=

F
z
GP
F
zi
GP
i 1=
n
∑
=
F
z
GS
F
zi
GS
i 1=
n
∑
=
F
y
GP
F
yi
GP
i 1=
n
∑
=

F
y
GS
F
yi
GS
i 1=
n
∑
=
M
y
GP
F
zi
GP
X
i
()
i 1=
n
∑
=
M
y
GS
F
zi
GS
X

i
()
i 1=
n
∑
=
M
z
GP
F
yi
GP
X
i
()
i 1=
n
∑
=